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A method for the determination of the spin and parity of a resonance through spin-rotation parameter measurements
I
I
Nuclear
Physics
(1969) 467-480.
North-Holland
Publ. Comp.,
Amsterdam
J
A METHOD FOR THE DETERMINATION OF THE SPIN AND PARITY OF A RESONANCE THROUGH SPIN-ROTATION PARAMETER MEASUREMENTS Nils A. TORNQVIST * CERN - Grncca
Received
23 I)ecember
1968
Abstract: A simple graphical method for studying polarization phenomena is presented. It provides a new method for the determination of the spin and parity of a KN- KN and KKresonance formed in the reactions nN -+nK, rrK+ K,\(C). + nh(x). This method of spin-parity determination requires measurements of spin-rotation parameters. and is based on the observation that the spin and parity of the dominating partial wave are simply related to the number of times and the when the scattering angle is varied direction the final polarization vector rotates. from 0 to 18Oo. This spin-parity determination would be independent of corresponding determinations using angular distributions. Apart from the spin-parity determination. the graphical technique is useful for a compact description of spin effects in terms of directly observable quantities. The method is applied and tested by plotting predictions of nN -+ nN phase-shift analysis.
1 INTRODUCTION The only directly observable quantities in a reaction AB - CD are the positive-definite cross sections as functions of energy, scattering angle and spin configuration of the particles a(E, 0, PA, PB, PC, PD), where A, B, C and D refer to other quantum numbers than those of space and spin, and P is the polarization vector. By using basic quantum mechanics and by imposing symmetries, theory furnishes the relations to express these cross sections in terms of complex amplitudes, not observed directly. In this paper a particularly simple graphical method is presented, which is very useful for studying the spin dependence in the particular case of spin(O-,i+) scattering. With this method, the relations between the observable cross sections and the underlying theory can easily be understood, and as an application it provides a new method for a direct and sensitive determination of the spin and parity of a resonance. The method involves graphs, which are projections in different planes of a three-dimensional diagram. Such a graph can be interpreted in three different ways: * On leave
from
Institute
of Suclear
Physics,
University
of Ilelsinki.
Ilelsinki.
468
K. A. TijRNQVIST
(i) As a plot of the final polarization vector projected on a particular plane. A special case is a plot of the spin rotation parameters A and R against each other or a plot of one of these against the polarization parameter ff. (ii) As a representation of the complex quantity Y = g/f (the ratio between the spin-flip and the non-spin-flip amplitude) on the complex sphere (Riemann sphere) in a particular projection. (iii) As graphs relating the observable quantities o(PB, PD)oo where u. is the differential cross section for an unpolarized target. By varying the energy or the scattering angle one obtains trajectories in the graphs. When the scattering angle is varied from 0 to 1800, the number of loops in the trajectory and the direction of these loops are uniquely related to the spin of the dominating partial wave. This property can be used for a direct determination of the spin and parity of a resonance. This method is independent of corresponding determinations from angular distributions. Although no spin-rotation data have yet been published which could be used for this kind of analysis, they can to-day, due to the advances in the technique of polarized targets, be measured with a remarkable accuracy. A group at CERN expects to achieve an accuracy of 10% in n+p scattering at high energies (p > 5 GeV/c) and low momentum transfers [l]. At lower energies it would be possible to measure in the whole angular interval because the energy of the scattered nucleon is always low. For other methods of spin and parity determination, see the review article in ref. [2]. Predictions of spin-rotation parameters from phase-shift analysis or Regge-pole theory have been studied by several authors [3-61. Many articles discussed polarized targets and their use can be found in ref. [7]. In sect. 2, we briefly summarize the basic formulas for spin effects in spin- (O-,f+) scattering, in order to define the notation and the conventions. In sect. 3, we describe the graphical method and discuss the spinparity determination. In sect. 4, we discuss the possible applications and test our method by studying predictions of phase-shift analysis in nN - nN scattering.
2. SPIN EFFECTS IN PSEUDOSCALAR SPIN- $+ BARYON SCATTERING The scattering ferent ways
MESON AND
matrix for spin- (O-, $+) scattering
IV = -A + iB ypqp
= .fl +f 2
o.kf
where A and B are the Lorentz-invariant amplitudes, and f and R the non-spin-flip ki spectively, qp is the four-momentum, tively, final c.m. momenta of the baryon,
o.ki
jkf i2
can be written in dif-
=f +igu.n,
amplitudes, fl and f2 the helicity and the spin-flip amplitude, reand kf are the initial, respec0 = (01, 02, 03) are the Pauli
469
SPIN-PARITYDETERMINATION matrices
and
is the normal
f+_ = f sin(+Q) -R cos(i8)
,
(2)
,
(3)
where 0 is the c.m. scattering angle. A convenient way to discuss polarization effects is in terms *of the spinspace density matrix. If the target baryon has a polarization P1, its density matrix is p’ = +(d+P’W)
(4)
,
then the density matrix for the final baryon is pf = MpiMt
= r2 oo(l +crPi.n){lt+Pf.o},
(5)
where u. is the differential cross section for an unpolarized target, cy is the ordinary polarization parameter, n is the normal to the scattering plane, and Pf is the polarization vector of the outgoing baryon cro = lfj2
+ (g12 ,
Q = 2 Im m*wJ, n = kixkf :kixkf( Pf(l+*Pi.n)=
(cr+Pi.n)n+pPix
03 (7)
,
(8)
’
n-y(Pixn)
xn
= {a+(1 -y)Pi*n)n+flPi
X n+yPi
(9)
,
where fi = Y
2 Re &*)/o,
,
={(f12 - IKI2jhJo
(10)
(11)
The polarization parameter CYis the easiest one to measure. This can be done either by measuring the differential cross section from a target polarized in the direction n, or by observing the polarization of the final baryon in the same direction. In order to measure the two other parameters P and y the target polarization must be in the scattering plane. Then we have for the final polarization vector Pf = cun+am+yl
,
(12)
where n, m and I are orthogonal vectors, m = I x n and I = Pi. Assuming for simplicity that the initial polarization is loo%, I, m and n form an orthonormal basis (the helicity frame of the initial baryon). The components of the polarization which are orthogonal to the momentum are the most readily measured quantities, while the parallel compo-
470
I\‘. A. TORNQVIST
nent would involve a spin-rotating magnet. However, both 3 and y can be measured as the orthogonal component if the experimental conditions are arranged so that the target polarization is parallel (9) or orthogonal (y) to the momentum of the scattered baryon. Experimentally more convenient quantities are the Wolfenstein spin-rotation parameters [3]
A = -3~0.~8 - ysin0 R = -3 sin0
+ y cos
,
(13)
0 ,
(14)
which are equal to the orthogonal component (in the scattering plane) of the final polarization, when the target polarization is parallel (A), respectively, orthogonal (R) to the beam momentum (see fig. 1). In terms of the amplitudes f ++ and f+_ (eqs. (2) and (3)) we have A = -2Re R =
(f++f+-*l/o,
(13’)
,
{1f++j2- ,f+-'ah,.
(14’)
Thus in the helicity frame of the final baryon the components zation vector are CY,A and R.
of the polari-
OirectDn t-4 measuring A,&r A,, if pi is turned 90?
Direction of andyring final potarisation (AWR)
Laboratory
Fig.
1. Illustration
of the kinematics
and the notation lab system.
in the c.m.
system
and in the
All the quantities above are defined in the c.m. system. They are in a non-relativistic approximation correct also in the laboratory if 8 is replaced by the lab scattering angle 8lab. The relativistically correct expressions for A and R are obtained [8] by replacing 0 by-0 +glab rather than by 0lab (the bar denotes the complement angle 1800-0 = 0). Thus relativistitally there appears an additional spin-rotating effect in the scattering plane 19, lo] by the angle 28lab- e. (This is related to the so-called Wick rotation.) The Wolfenstein parameters for the recoil baryon in the lab system are thus [ll]
- A ret = -$cOs (@-olab) + rsin(8-alab) R ret = p sin (e-el,b)
- + y cos (e-olab)
,
(15)
.
(16)
SPlX-PARITY
IIIETERMMATIOS
471
Because
of conventional reasons there is a change of sign in the definition and &ec compared to A and R. Of Arec Observe that Q, 3 and y are not independent but satisfy cY2+,32+y2 = 1 )
(17)
and similarly $+A2+@ Q~.A~,~~+R;~~
= 1,
(18)
= 1 .
(19)
Thus measurement of cy and one of the spin-rotation parameters determines the spin dependence completely up to a sign of A or R. If the spin dependence is completely known experimentally then f and g can be determined up to a common phase, which at least in principle would determine the phase shifts unambiguously. Finally a useful quantity is the spin-rotation angle 6 = arctg:
= arctg:
3. THE GRAPHS AND THEIR APPLICATION PARITY DETERMINATION
- 0.
FOR SPIN AND
In the following we shall assume that the initial polarization vector Pi is in the scattering plane, and we shall always be working in the c.m. system. Let us represent the final polarization vector pf, (12), as the radius vector in the three-dimensional space spanned by the basic vectors I, m and n. Then due to eq. (17), Pf lies on the unit sphere. Varying the scattering angle (or the energy), pf describes trajectories on this sphere. The components of Pf are (Y, J and y. See fig. 2. The projection of the threedimensional trajectory on any plane is a curve which is confined to be in-
-hi
i
Y
a
b
Fig. 2. The final polarization vector visualixcd in a three-dimensional coordinate system and the projection in the P-y plane. The curve shows a possible trajectory obtained when varying the scattering angle from 0 to 18Oo.
472
N.A.TtiRNQV1S.T
side the unit circle. The trajectories must form closed loops, since for 0 = 0 and 180°, the spin-flip amplitude vanishes. The curves thus begin and end at the point (O,O, 1). Exceptionally this need not be true, if also f accidentally vanishes in forward or backward scattering, i.e., there is a dip in the differential cross section. In terms of the amplitudes f and g, we can give another interesting interpretation of the three-dimensional graph. From the definition of Q, ,3 and Y, 21m y Q= _~ l+ (Y12'
i3= 2Rey l+ (YW
(21)
where r = g/f, it follows directly [12] that we can interpret the three-dimensional diagram as the representation of the complex number Y on the complex sphere (Riemann sphere). The origin Y = 0 is represented by the ‘north pole’ (O,O, l), the unit circle 1Y) = 1 by the ‘equator’ (a, /3,0), and the infinity 1YI = ~0 by the ‘south pole’ (O,O, -1). It is also of some interest to note that a parallel projection of the threedimensional graph on a plane can be interpreted as the complex plane for a quantity simply related to f and g. Thus for example, the projections on the coordinate planes (Q, 8,0), ((Y,0, y) and (0,9, y) correspond to a+@
= 2ifg*/ao
,
a+ iy = i(f+d,(f-R3*/oo 9 + + = i(f-ig)(f+ig)*/oo Note also that an SU2 transformation
of the vector
(22) (23)
, .
(24) (0 corresponds
to an
03 rotation of the vector
F and a set of homographic transformations on 0 r [r’ = (ar+br)/(cy+dr)]. This just expresses the well-known correspondences between these groups of transformations. Finally we note that the cross section for a process where the initial polarization is in the scattering plane and the final polarization is analyzed in the direction M is oM
=
If
eeii”M =
cos(+B~)
+ ge
+iiV511sin
(+ eM)
12
$uo(l+asincP/M sine/M +!?lcos(p/M sinO/M
+yCOS~,j,f) .
(25)
Here 6,$,f and q,+,, are the spherical angles of the direction M in a reference frame where Pi is the z-axis and eM is the angle between Pi and M, and q,+f is the angle between M and the scattering plane. The quantities YM = 20M/oo are graphically represented as the distance from a tangent plane of the sphere to the point on the sphere given by f and g. The tangent plane is given by the point of intersection between the direction -M and the sphere. This shows clearly the close relationship between;.points in the graph and measurable quantities, and how the continuum of observable cross sections (one for every BM and ‘P/~) are related. For example if the
SPIN-PARITY
473
DETERMINATION
the final polarization is analyzed in the scattering plane, then in a 3-y plot (see fig. 2), the distance from the tangent to the point on the curve * is the quantity yM . Determination of sfiin and parily . We assume that one partial wave dominates and neglect the others at the moment. Then from the partial-wave expansion for f andg
f(O) = x T{(z+l)nl+ g(O) = X
C
+&-)pl(cos
(al+
-al_)P$cos
0) ,
0) ,
1 we see that the ratio r(O) = K(e)/!(O)
pj(c0se)
1
rz-@) = -
A 6n-
is
1
pl(cosq
9
ifj = .l+$ ,
(28)
if j = 1-i .
(29)
lf w
5X-
4n-
3n-
0
a
f
Fig. 3. An example of the spin-rotation eq. (30), when dominance of one partial
f
angle as a function wave is assumed. j
=
of the scattering angle, The example is for 1=3.
i.
* When the graphs are read in this way, one can see the relationship to the graphs for studying isospin invariance [ 131, where branching ratios for different charge channels of a reaction are plotted against each other.
This is a real quantity from which it follows that the polarization eter cy is zero, while the spin-rotation angle is
w*(e)
@z,(e)= arc@
= 2 arc tg rZ,(G) .
param-
(30)
1 - (yz*PP
These functions are very smooth and monotonous functions of the scattering angle (they are in fact well approximated by the linear functions *2Z0, see the example in fig. 3, where &3+ is displayed). They have the interesting property that they vary from 0 to +2nZ when 8 is varied from 0 to II. Thus if there is no background from other partial waves, the curves in the P-r plot form 2 loops along the unit circle. In fig. 4 to the left we indicate the behaviour of the curves for the lowest 1. The arrows indicate the direction of increasing 0. Let N be the number of loops and let the sign of N be positive if the rotation is anticlockwise for increasing 0 and otherwise negative. Then the orbital angular momentum 1, the total angular monentum j and the parity P of the partial wave are I= j=
pq N+$
(31)
, t
(32)
.
(33)
P = -(-l)N
Similar results are obtained in a plot of A against R (see fig. 4 to the right). Then there is an additional rotation by n (and a reflection due to sign conventions), whereby 1 = IN+f 1, j = JArI and P = -(-l)“+f. When spin-rotation data become available, these properties can be used for a direct determination of the spin and parity of a resonance. This determination is experimentally completely independent of spin determinations from angular distributions. Of course, generally, there is always a background from other partial waves. However, if the background is not too large, we can expect that it mainly has the effect of moving the loops from the boundary inside the circle and distorting their circular shape. In sect. 4 we shall find that this indeed is the case in nN + nN. If the background is large, the picture will of course be obscured and one must use phase-shift analysis to entangle the resonances. A good illustration of the importance of such interference effects is the following rather amusing effect. We assume ideal parity doubling, i.e., that only two equally large partial waves of same j but opposite parity contribute. Then it follows from the properties of the Legendre polynomials that for any j = Z+q P:,l(COS
y(e) = z+
0) - Pj(cos
0)
P~+~(COS 0)+ ~~(~0s e) = cot($e)
,
(34)
and @J(O)= -0 . The curve in the b-y plot is the lower half-circle
(35) (see fig. 4f). It does not
SPIN-PARITY
LSNI,J:
475
DETERMINATION
lN*t/ZI.P=-(-1):
N=-312
N..l
R
.Y
.~
--.--
e)
L.2
N=+2 Y
N.-S/2
J’5l2
Fig. 4. The expected behaviour of the curves in a P-y plot (to the left) and an A-R plot (to the right) when assuming dominance of one partial wave (4a-e). Fig. 4f shows the behaviour in the case of ideal parity doubling.
have to form a loop because in backward scattering the two partial waves cancel so that both f and R are zero. In an A-R plot, one would obtain just a point R = 1, A = 0. Thus for ideal parity doubling the final helicity is the same as the initial, i.e., f+_ = 0. This result could of course have been obtained directly, since f+_ is proportional to the difference between the two amplitudes of opposite parity. Thus interference effects are important, but we see from the preceding discussion that spin-rotation data are also very sensitive to the quantum numbers of an individual partial wave, while the polarization, LY,only reflects interference effects *. * This is related
to the fact that spin rotation solving ambiguities in phase-shift analyses ent solutions give very different predictions
measurements arc very useful (4.51. In the 3-y or R-A plots, to the curves.
for rediffer-
476
N. A. TtiRNQVIST
Finally we should point out that we have assumed the validity of the conventional formalism. If, for example, parity conservation would not be strictly valid, there would be four invariant amplitudes and the conclusions made above would not hold. Obviously spin-rotation measurements are a very good test for the over-all correctness of the conventional formalism. 4. POSSIBLE APPLICATIONS. PREDICTIONS OF PHASE-SHIFT ANALYSIS IN nN - nN SCATTERING The reactions for which our type of analysis is applicable are The exnN-KA, nN-KKC, KN-KN, KN-rnCandKN-7rh. TN -TN, periments must use a polarized target and the polarization of the outgoing baryon must be analyzed. In case the final baryon is a strange particle, especially, the polarization can be fairly easily analyzed by observing the decay asymmetry of the baryon. In the case of elastic scattering the final polarization must be analyzed by a rescattering. At present no such data have been published. But this type of experiments are being planned and performed, e.g., at CERN where spin-rotation parameters are being measured in nN - nN by a group from Saclay. They hope [l] to achieve an accuracy as high as 10% which is more than sufficient for our type of analysis. This experiment is aimed at high energies (5-18 GeV/c) and low-momentum transfers in order to test Regge-pole theory predictions. At lower energies, in the resonance region, one could measure in the whole angular interval since the energy of the recoil nucleon is always low and the separation of inelastic events easier. However, one does not necessarily need data in the whole angular region for the spin-parity determination. In fact, only two or three points should in principle be enough to distinguish between the different cases in fig. 4. Another fact, whichsimplifies the experiments, is that the normalization of the cross sections is unimportant. In the following we shall see what nN - nN phase-shifts analysis predicts in the energy regions of the lowest resonances. We use the phase-shift
Fig. 5. (a) Thefl-y and (b) the a-yplot for ~‘p -7r+p in the energy region of the h(1236). with varying scattering angles as predicted by the CERN phase-shift analysis. The numbers along the curves are the cosine of the c.m. scattering angle.
SPIN-PARITY
477
DETERMINATION
analysis done by Lovelace et al. [14] at CEHN. In fig. 5, the P-7 and (Y-Y plots are shown at IX,.,. = 1230 MeV for n+p - n+p. Here the ~(1236) dominates and as expected the curve in the /3-y plot describes one anticlockwise loop, which corresponds to a P-wave and jP = 3’. The background from other partial waves gives rise to a small cy, which can be seen in the d-y plot. The next resonance N(1470) is in the T = f channel. We show the predicted /3-y plots both for the pure T = f and for the n-p - a-p channel. The n-p + 71-p contains some (one-third) of the T =4 amplitude, but it is difficult to experimentally separate the T = $ process since it would require spinrotation measurements for the charge exchange process as well as for the elastic scattering. We see, however, in fig. 6, that in both the T = f and in the n-p -+ n-p, there is again one loop but in the opposite direction to that of the previous case. Thus we quite correctly get 1 = 1 and jP = f’.
Fig. 6. The P-7 plots for the pure T = & channel and the n-p = n-p channel in the energy region of the N(1470) with varying scattering angle as predicted by the CERK phase-shift analysis. u
E_.su
Fig.
7. Same as in fig.
6
L,,.
Ԧ4W
P,_.
777
but in the energy
u
.P_.P
__ ... . . t__,
region
of the N(1518).
478
N. A. TijRNQVIST
The predictions in the following resonance region N(1518) are shown in fig. ‘7. Here there is already considerable background due to other partial waves. We can nevertheless still see the two clockwise loops, which should be expected from a jp = #- resonance. = 1680 MeV, there are two nearby T = $ resonances of Around EC.,. same j = 4, but opposite parity N(1680) and N(1688). Thus strong interference effects are expected. From the phase-shift analysis predictions in fig. 8, we see that in backward scattering (especially in the pure T = i channel) the curves behave very much as should be expected from parity doubling (cf. fig. 4f). Figs. 9 and 10 show the predictions in the regions of the A(1950) and the N(2190). Due to the huge background - the latest CERN phase-shift analysis
Fig. 8. Same as in fig. 6. but in the energy regions resonances.
of the N(1680) and the N(1688)
Fig. 9. Same as in fig. 5a. but in the energy
region
of the A(l950).
SPIIi-PARITY
Fig.
10.
Same
as in fig.
6,
DETERMINATION
but in the energy
region
479
of the
N(2190).
predicts several new, although not well established, resonances in this region - and due to the large inelasticity of the resonances, the ‘signal-tonoise ratio’ is very small. The curves can then behave in an unpredictable manner. The fact that the curve in fig. 10 forms four clockwise loops (although the fourth is very small), which quite correctly corresponds to the quantum numbers jp= i- of the N(2190), must therefore be considered partly as an accident. There is, however, much structure, which shows that there is a great amount of information to be obtained from spin-rotation data.
5. CONCLUSIONS We have shown that spin-rotation data can be used for a direct and independent determination of the spin and parity of a resonance if the background from other partial waves is not too large. The method is described using a simple graphical technique, by which the spin and parity are related to loops in a plot of spin-rotation parameters against each others. Apart from the spin and parity determination, the graphical method is a useful tool to study spin effects phenomenologically, since points in the diagram are simply related both to directly observable cross sections and to the spinflip and spin-non-flip amplitudes. When the method is tested by using predictions from pion-nucleon phaseshift analysis to spin-rotation parameters, the curves behave as expected from the known spins and parities of the low-mass resonances. In the case of the higher-mass resonances, the loops are distorted by interference effects from other large partial waves. However, if the background is not very large, the expected loops are mainly distorted in their shape and position, while their number and their direction, which are the properties which determine the spin and parity, are left intact.
480
N. A. TGRNQVIST
Many thanks are due to Professor R. Armenteros, Professor J. Prentki, Dr. M. Roos and Dr. L. Van Rossum for discussions. I am also grateful to Professor J. Prentki and Professor W. Thirring for the kind hospitality at the Theoretical Study Division at CEF?,N.
REFERENCES [l] [2] [3] [4] [5] [6]
L. Van Rossum, private communication. R. D. Tripp, Ann. Rev. Nucl. Sci. 15 (1965) 325. L. Wolfenstein, Phys. Rev. 96 (1954) 1654; Ann. Rev. Nucl. Sci. 6 (1956) 43. Y.S.Kim, Phys. Rev. 129 (1963) 862. L. D. Roper, Phys. Rev. 155 (1967) 1744. W. Rarita, R. J. Riddel, C. B. Chiu and R. J. N. Phillips, UCRL-17523 (1967)) unpublished. [7] Proc. Int. Conf. on polarized targets and ion sources, Saclay (1966). [8] H. P. Stapp, Phys. Rev. 103 (1956) 425. [9] G. C. Wick, Ann. of Phys. 18 (1962) 65. [lo] A. Biahs and B. E. Y. Svensson, Nuovo Cimento 42 (1965) 908. [ll] H. Steiner, UCRL-17903 (1967), unpublished. [12] P. du Val, Homographies , quaternions and rotations (Oxford University Press, 1964). 113) X. A. Tijrnqvist, Phys. Rev. 161 (1967) 1581; Nucl. Phys. B6 (1968) 187. [14] C. Lovelace, Proc. Heidelberg Int. Conf. on elementary particles, 1967 (NorthHolland Publ. Comp., Amsterdam, 1968); A. Donnachie , R. G. Kirsopp and C. Lovelace, Phys. Letters 26B (1968) 161.
I
Nuclear
Physics
(1969) 467-480.
North-Holland
Publ. Comp.,
Amsterdam
J
A METHOD FOR THE DETERMINATION OF THE SPIN AND PARITY OF A RESONANCE THROUGH SPIN-ROTATION PARAMETER MEASUREMENTS Nils A. TORNQVIST * CERN - Grncca
Received
23 I)ecember
1968
Abstract: A simple graphical method for studying polarization phenomena is presented. It provides a new method for the determination of the spin and parity of a KN- KN and KKresonance formed in the reactions nN -+nK, rrK+ K,\(C). + nh(x). This method of spin-parity determination requires measurements of spin-rotation parameters. and is based on the observation that the spin and parity of the dominating partial wave are simply related to the number of times and the when the scattering angle is varied direction the final polarization vector rotates. from 0 to 18Oo. This spin-parity determination would be independent of corresponding determinations using angular distributions. Apart from the spin-parity determination. the graphical technique is useful for a compact description of spin effects in terms of directly observable quantities. The method is applied and tested by plotting predictions of nN -+ nN phase-shift analysis.
1 INTRODUCTION The only directly observable quantities in a reaction AB - CD are the positive-definite cross sections as functions of energy, scattering angle and spin configuration of the particles a(E, 0, PA, PB, PC, PD), where A, B, C and D refer to other quantum numbers than those of space and spin, and P is the polarization vector. By using basic quantum mechanics and by imposing symmetries, theory furnishes the relations to express these cross sections in terms of complex amplitudes, not observed directly. In this paper a particularly simple graphical method is presented, which is very useful for studying the spin dependence in the particular case of spin(O-,i+) scattering. With this method, the relations between the observable cross sections and the underlying theory can easily be understood, and as an application it provides a new method for a direct and sensitive determination of the spin and parity of a resonance. The method involves graphs, which are projections in different planes of a three-dimensional diagram. Such a graph can be interpreted in three different ways: * On leave
from
Institute
of Suclear
Physics,
University
of Ilelsinki.
Ilelsinki.
468
K. A. TijRNQVIST
(i) As a plot of the final polarization vector projected on a particular plane. A special case is a plot of the spin rotation parameters A and R against each other or a plot of one of these against the polarization parameter ff. (ii) As a representation of the complex quantity Y = g/f (the ratio between the spin-flip and the non-spin-flip amplitude) on the complex sphere (Riemann sphere) in a particular projection. (iii) As graphs relating the observable quantities o(PB, PD)oo where u. is the differential cross section for an unpolarized target. By varying the energy or the scattering angle one obtains trajectories in the graphs. When the scattering angle is varied from 0 to 1800, the number of loops in the trajectory and the direction of these loops are uniquely related to the spin of the dominating partial wave. This property can be used for a direct determination of the spin and parity of a resonance. This method is independent of corresponding determinations from angular distributions. Although no spin-rotation data have yet been published which could be used for this kind of analysis, they can to-day, due to the advances in the technique of polarized targets, be measured with a remarkable accuracy. A group at CERN expects to achieve an accuracy of 10% in n+p scattering at high energies (p > 5 GeV/c) and low momentum transfers [l]. At lower energies it would be possible to measure in the whole angular interval because the energy of the scattered nucleon is always low. For other methods of spin and parity determination, see the review article in ref. [2]. Predictions of spin-rotation parameters from phase-shift analysis or Regge-pole theory have been studied by several authors [3-61. Many articles discussed polarized targets and their use can be found in ref. [7]. In sect. 2, we briefly summarize the basic formulas for spin effects in spin- (O-,f+) scattering, in order to define the notation and the conventions. In sect. 3, we describe the graphical method and discuss the spinparity determination. In sect. 4, we discuss the possible applications and test our method by studying predictions of phase-shift analysis in nN - nN scattering.
2. SPIN EFFECTS IN PSEUDOSCALAR SPIN- $+ BARYON SCATTERING The scattering ferent ways
MESON AND
matrix for spin- (O-, $+) scattering
IV = -A + iB ypqp
= .fl +f 2
o.kf
where A and B are the Lorentz-invariant amplitudes, and f and R the non-spin-flip ki spectively, qp is the four-momentum, tively, final c.m. momenta of the baryon,
o.ki
jkf i2
can be written in dif-
=f +igu.n,
amplitudes, fl and f2 the helicity and the spin-flip amplitude, reand kf are the initial, respec0 = (01, 02, 03) are the Pauli
469
SPIN-PARITYDETERMINATION matrices
and
is the normal
f+_ = f sin(+Q) -R cos(i8)
,
(2)
,
(3)
where 0 is the c.m. scattering angle. A convenient way to discuss polarization effects is in terms *of the spinspace density matrix. If the target baryon has a polarization P1, its density matrix is p’ = +(d+P’W)
(4)
,
then the density matrix for the final baryon is pf = MpiMt
= r2 oo(l +crPi.n){lt+Pf.o},
(5)
where u. is the differential cross section for an unpolarized target, cy is the ordinary polarization parameter, n is the normal to the scattering plane, and Pf is the polarization vector of the outgoing baryon cro = lfj2
+ (g12 ,
Q = 2 Im m*wJ, n = kixkf :kixkf( Pf(l+*Pi.n)=
(cr+Pi.n)n+pPix
03 (7)
,
(8)
’
n-y(Pixn)
xn
= {a+(1 -y)Pi*n)n+flPi
X n+yPi
(9)
,
where fi = Y
2 Re &*)/o,
,
={(f12 - IKI2jhJo
(10)
(11)
The polarization parameter CYis the easiest one to measure. This can be done either by measuring the differential cross section from a target polarized in the direction n, or by observing the polarization of the final baryon in the same direction. In order to measure the two other parameters P and y the target polarization must be in the scattering plane. Then we have for the final polarization vector Pf = cun+am+yl
,
(12)
where n, m and I are orthogonal vectors, m = I x n and I = Pi. Assuming for simplicity that the initial polarization is loo%, I, m and n form an orthonormal basis (the helicity frame of the initial baryon). The components of the polarization which are orthogonal to the momentum are the most readily measured quantities, while the parallel compo-
470
I\‘. A. TORNQVIST
nent would involve a spin-rotating magnet. However, both 3 and y can be measured as the orthogonal component if the experimental conditions are arranged so that the target polarization is parallel (9) or orthogonal (y) to the momentum of the scattered baryon. Experimentally more convenient quantities are the Wolfenstein spin-rotation parameters [3]
A = -3~0.~8 - ysin0 R = -3 sin0
+ y cos
,
(13)
0 ,
(14)
which are equal to the orthogonal component (in the scattering plane) of the final polarization, when the target polarization is parallel (A), respectively, orthogonal (R) to the beam momentum (see fig. 1). In terms of the amplitudes f ++ and f+_ (eqs. (2) and (3)) we have A = -2Re R =
(f++f+-*l/o,
(13’)
,
{1f++j2- ,f+-'ah,.
(14’)
Thus in the helicity frame of the final baryon the components zation vector are CY,A and R.
of the polari-
OirectDn t-4 measuring A,&r A,, if pi is turned 90?
Direction of andyring final potarisation (AWR)
Laboratory
Fig.
1. Illustration
of the kinematics
and the notation lab system.
in the c.m.
system
and in the
All the quantities above are defined in the c.m. system. They are in a non-relativistic approximation correct also in the laboratory if 8 is replaced by the lab scattering angle 8lab. The relativistically correct expressions for A and R are obtained [8] by replacing 0 by-0 +glab rather than by 0lab (the bar denotes the complement angle 1800-0 = 0). Thus relativistitally there appears an additional spin-rotating effect in the scattering plane 19, lo] by the angle 28lab- e. (This is related to the so-called Wick rotation.) The Wolfenstein parameters for the recoil baryon in the lab system are thus [ll]
- A ret = -$cOs (@-olab) + rsin(8-alab) R ret = p sin (e-el,b)
- + y cos (e-olab)
,
(15)
.
(16)
SPlX-PARITY
IIIETERMMATIOS
471
Because
of conventional reasons there is a change of sign in the definition and &ec compared to A and R. Of Arec Observe that Q, 3 and y are not independent but satisfy cY2+,32+y2 = 1 )
(17)
and similarly $+A2+@ Q~.A~,~~+R;~~
= 1,
(18)
= 1 .
(19)
Thus measurement of cy and one of the spin-rotation parameters determines the spin dependence completely up to a sign of A or R. If the spin dependence is completely known experimentally then f and g can be determined up to a common phase, which at least in principle would determine the phase shifts unambiguously. Finally a useful quantity is the spin-rotation angle 6 = arctg:
= arctg:
3. THE GRAPHS AND THEIR APPLICATION PARITY DETERMINATION
- 0.
FOR SPIN AND
In the following we shall assume that the initial polarization vector Pi is in the scattering plane, and we shall always be working in the c.m. system. Let us represent the final polarization vector pf, (12), as the radius vector in the three-dimensional space spanned by the basic vectors I, m and n. Then due to eq. (17), Pf lies on the unit sphere. Varying the scattering angle (or the energy), pf describes trajectories on this sphere. The components of Pf are (Y, J and y. See fig. 2. The projection of the threedimensional trajectory on any plane is a curve which is confined to be in-
-hi
i
Y
a
b
Fig. 2. The final polarization vector visualixcd in a three-dimensional coordinate system and the projection in the P-y plane. The curve shows a possible trajectory obtained when varying the scattering angle from 0 to 18Oo.
472
N.A.TtiRNQV1S.T
side the unit circle. The trajectories must form closed loops, since for 0 = 0 and 180°, the spin-flip amplitude vanishes. The curves thus begin and end at the point (O,O, 1). Exceptionally this need not be true, if also f accidentally vanishes in forward or backward scattering, i.e., there is a dip in the differential cross section. In terms of the amplitudes f and g, we can give another interesting interpretation of the three-dimensional graph. From the definition of Q, ,3 and Y, 21m y Q= _~ l+ (Y12'
i3= 2Rey l+ (YW
(21)
where r = g/f, it follows directly [12] that we can interpret the three-dimensional diagram as the representation of the complex number Y on the complex sphere (Riemann sphere). The origin Y = 0 is represented by the ‘north pole’ (O,O, l), the unit circle 1Y) = 1 by the ‘equator’ (a, /3,0), and the infinity 1YI = ~0 by the ‘south pole’ (O,O, -1). It is also of some interest to note that a parallel projection of the threedimensional graph on a plane can be interpreted as the complex plane for a quantity simply related to f and g. Thus for example, the projections on the coordinate planes (Q, 8,0), ((Y,0, y) and (0,9, y) correspond to a+@
= 2ifg*/ao
,
a+ iy = i(f+d,(f-R3*/oo 9 + + = i(f-ig)(f+ig)*/oo Note also that an SU2 transformation
of the vector
(22) (23)
, .
(24) (0 corresponds
to an
03 rotation of the vector
F and a set of homographic transformations on 0 r [r’ = (ar+br)/(cy+dr)]. This just expresses the well-known correspondences between these groups of transformations. Finally we note that the cross section for a process where the initial polarization is in the scattering plane and the final polarization is analyzed in the direction M is oM
=
If
eeii”M =
cos(+B~)
+ ge
+iiV511sin
(+ eM)
12
$uo(l+asincP/M sine/M +!?lcos(p/M sinO/M
+yCOS~,j,f) .
(25)
Here 6,$,f and q,+,, are the spherical angles of the direction M in a reference frame where Pi is the z-axis and eM is the angle between Pi and M, and q,+f is the angle between M and the scattering plane. The quantities YM = 20M/oo are graphically represented as the distance from a tangent plane of the sphere to the point on the sphere given by f and g. The tangent plane is given by the point of intersection between the direction -M and the sphere. This shows clearly the close relationship between;.points in the graph and measurable quantities, and how the continuum of observable cross sections (one for every BM and ‘P/~) are related. For example if the
SPIN-PARITY
473
DETERMINATION
the final polarization is analyzed in the scattering plane, then in a 3-y plot (see fig. 2), the distance from the tangent to the point on the curve * is the quantity yM . Determination of sfiin and parily . We assume that one partial wave dominates and neglect the others at the moment. Then from the partial-wave expansion for f andg
f(O) = x T{(z+l)nl+ g(O) = X
C
+&-)pl(cos
(al+
-al_)P$cos
0) ,
0) ,
1 we see that the ratio r(O) = K(e)/!(O)
pj(c0se)
1
rz-@) = -
A 6n-
is
1
pl(cosq
9
ifj = .l+$ ,
(28)
if j = 1-i .
(29)
lf w
5X-
4n-
3n-
0
a
f
Fig. 3. An example of the spin-rotation eq. (30), when dominance of one partial
f
angle as a function wave is assumed. j
=
of the scattering angle, The example is for 1=3.
i.
* When the graphs are read in this way, one can see the relationship to the graphs for studying isospin invariance [ 131, where branching ratios for different charge channels of a reaction are plotted against each other.
This is a real quantity from which it follows that the polarization eter cy is zero, while the spin-rotation angle is
w*(e)
@z,(e)= arc@
= 2 arc tg rZ,(G) .
param-
(30)
1 - (yz*PP
These functions are very smooth and monotonous functions of the scattering angle (they are in fact well approximated by the linear functions *2Z0, see the example in fig. 3, where &3+ is displayed). They have the interesting property that they vary from 0 to +2nZ when 8 is varied from 0 to II. Thus if there is no background from other partial waves, the curves in the P-r plot form 2 loops along the unit circle. In fig. 4 to the left we indicate the behaviour of the curves for the lowest 1. The arrows indicate the direction of increasing 0. Let N be the number of loops and let the sign of N be positive if the rotation is anticlockwise for increasing 0 and otherwise negative. Then the orbital angular momentum 1, the total angular monentum j and the parity P of the partial wave are I= j=
pq N+$
(31)
, t
(32)
.
(33)
P = -(-l)N
Similar results are obtained in a plot of A against R (see fig. 4 to the right). Then there is an additional rotation by n (and a reflection due to sign conventions), whereby 1 = IN+f 1, j = JArI and P = -(-l)“+f. When spin-rotation data become available, these properties can be used for a direct determination of the spin and parity of a resonance. This determination is experimentally completely independent of spin determinations from angular distributions. Of course, generally, there is always a background from other partial waves. However, if the background is not too large, we can expect that it mainly has the effect of moving the loops from the boundary inside the circle and distorting their circular shape. In sect. 4 we shall find that this indeed is the case in nN + nN. If the background is large, the picture will of course be obscured and one must use phase-shift analysis to entangle the resonances. A good illustration of the importance of such interference effects is the following rather amusing effect. We assume ideal parity doubling, i.e., that only two equally large partial waves of same j but opposite parity contribute. Then it follows from the properties of the Legendre polynomials that for any j = Z+q P:,l(COS
y(e) = z+
0) - Pj(cos
0)
P~+~(COS 0)+ ~~(~0s e) = cot($e)
,
(34)
and @J(O)= -0 . The curve in the b-y plot is the lower half-circle
(35) (see fig. 4f). It does not
SPIN-PARITY
LSNI,J:
475
DETERMINATION
lN*t/ZI.P=-(-1):
N=-312
N..l
R
.Y
.~
--.--
e)
L.2
N=+2 Y
N.-S/2
J’5l2
Fig. 4. The expected behaviour of the curves in a P-y plot (to the left) and an A-R plot (to the right) when assuming dominance of one partial wave (4a-e). Fig. 4f shows the behaviour in the case of ideal parity doubling.
have to form a loop because in backward scattering the two partial waves cancel so that both f and R are zero. In an A-R plot, one would obtain just a point R = 1, A = 0. Thus for ideal parity doubling the final helicity is the same as the initial, i.e., f+_ = 0. This result could of course have been obtained directly, since f+_ is proportional to the difference between the two amplitudes of opposite parity. Thus interference effects are important, but we see from the preceding discussion that spin-rotation data are also very sensitive to the quantum numbers of an individual partial wave, while the polarization, LY,only reflects interference effects *. * This is related
to the fact that spin rotation solving ambiguities in phase-shift analyses ent solutions give very different predictions
measurements arc very useful (4.51. In the 3-y or R-A plots, to the curves.
for rediffer-
476
N. A. TtiRNQVIST
Finally we should point out that we have assumed the validity of the conventional formalism. If, for example, parity conservation would not be strictly valid, there would be four invariant amplitudes and the conclusions made above would not hold. Obviously spin-rotation measurements are a very good test for the over-all correctness of the conventional formalism. 4. POSSIBLE APPLICATIONS. PREDICTIONS OF PHASE-SHIFT ANALYSIS IN nN - nN SCATTERING The reactions for which our type of analysis is applicable are The exnN-KA, nN-KKC, KN-KN, KN-rnCandKN-7rh. TN -TN, periments must use a polarized target and the polarization of the outgoing baryon must be analyzed. In case the final baryon is a strange particle, especially, the polarization can be fairly easily analyzed by observing the decay asymmetry of the baryon. In the case of elastic scattering the final polarization must be analyzed by a rescattering. At present no such data have been published. But this type of experiments are being planned and performed, e.g., at CERN where spin-rotation parameters are being measured in nN - nN by a group from Saclay. They hope [l] to achieve an accuracy as high as 10% which is more than sufficient for our type of analysis. This experiment is aimed at high energies (5-18 GeV/c) and low-momentum transfers in order to test Regge-pole theory predictions. At lower energies, in the resonance region, one could measure in the whole angular interval since the energy of the recoil nucleon is always low and the separation of inelastic events easier. However, one does not necessarily need data in the whole angular region for the spin-parity determination. In fact, only two or three points should in principle be enough to distinguish between the different cases in fig. 4. Another fact, whichsimplifies the experiments, is that the normalization of the cross sections is unimportant. In the following we shall see what nN - nN phase-shifts analysis predicts in the energy regions of the lowest resonances. We use the phase-shift
Fig. 5. (a) Thefl-y and (b) the a-yplot for ~‘p -7r+p in the energy region of the h(1236). with varying scattering angles as predicted by the CERN phase-shift analysis. The numbers along the curves are the cosine of the c.m. scattering angle.
SPIN-PARITY
477
DETERMINATION
analysis done by Lovelace et al. [14] at CEHN. In fig. 5, the P-7 and (Y-Y plots are shown at IX,.,. = 1230 MeV for n+p - n+p. Here the ~(1236) dominates and as expected the curve in the /3-y plot describes one anticlockwise loop, which corresponds to a P-wave and jP = 3’. The background from other partial waves gives rise to a small cy, which can be seen in the d-y plot. The next resonance N(1470) is in the T = f channel. We show the predicted /3-y plots both for the pure T = f and for the n-p - a-p channel. The n-p + 71-p contains some (one-third) of the T =4 amplitude, but it is difficult to experimentally separate the T = $ process since it would require spinrotation measurements for the charge exchange process as well as for the elastic scattering. We see, however, in fig. 6, that in both the T = f and in the n-p -+ n-p, there is again one loop but in the opposite direction to that of the previous case. Thus we quite correctly get 1 = 1 and jP = f’.
Fig. 6. The P-7 plots for the pure T = & channel and the n-p = n-p channel in the energy region of the N(1470) with varying scattering angle as predicted by the CERK phase-shift analysis. u
E_.su
Fig.
7. Same as in fig.
6
L,,.
Ԧ4W
P,_.
777
but in the energy
u
.P_.P
__ ... . . t__,
region
of the N(1518).
478
N. A. TijRNQVIST
The predictions in the following resonance region N(1518) are shown in fig. ‘7. Here there is already considerable background due to other partial waves. We can nevertheless still see the two clockwise loops, which should be expected from a jp = #- resonance. = 1680 MeV, there are two nearby T = $ resonances of Around EC.,. same j = 4, but opposite parity N(1680) and N(1688). Thus strong interference effects are expected. From the phase-shift analysis predictions in fig. 8, we see that in backward scattering (especially in the pure T = i channel) the curves behave very much as should be expected from parity doubling (cf. fig. 4f). Figs. 9 and 10 show the predictions in the regions of the A(1950) and the N(2190). Due to the huge background - the latest CERN phase-shift analysis
Fig. 8. Same as in fig. 6. but in the energy regions resonances.
of the N(1680) and the N(1688)
Fig. 9. Same as in fig. 5a. but in the energy
region
of the A(l950).
SPIIi-PARITY
Fig.
10.
Same
as in fig.
6,
DETERMINATION
but in the energy
region
479
of the
N(2190).
predicts several new, although not well established, resonances in this region - and due to the large inelasticity of the resonances, the ‘signal-tonoise ratio’ is very small. The curves can then behave in an unpredictable manner. The fact that the curve in fig. 10 forms four clockwise loops (although the fourth is very small), which quite correctly corresponds to the quantum numbers jp= i- of the N(2190), must therefore be considered partly as an accident. There is, however, much structure, which shows that there is a great amount of information to be obtained from spin-rotation data.
5. CONCLUSIONS We have shown that spin-rotation data can be used for a direct and independent determination of the spin and parity of a resonance if the background from other partial waves is not too large. The method is described using a simple graphical technique, by which the spin and parity are related to loops in a plot of spin-rotation parameters against each others. Apart from the spin and parity determination, the graphical method is a useful tool to study spin effects phenomenologically, since points in the diagram are simply related both to directly observable cross sections and to the spinflip and spin-non-flip amplitudes. When the method is tested by using predictions from pion-nucleon phaseshift analysis to spin-rotation parameters, the curves behave as expected from the known spins and parities of the low-mass resonances. In the case of the higher-mass resonances, the loops are distorted by interference effects from other large partial waves. However, if the background is not very large, the expected loops are mainly distorted in their shape and position, while their number and their direction, which are the properties which determine the spin and parity, are left intact.
480
N. A. TGRNQVIST
Many thanks are due to Professor R. Armenteros, Professor J. Prentki, Dr. M. Roos and Dr. L. Van Rossum for discussions. I am also grateful to Professor J. Prentki and Professor W. Thirring for the kind hospitality at the Theoretical Study Division at CEF?,N.
REFERENCES [l] [2] [3] [4] [5] [6]
L. Van Rossum, private communication. R. D. Tripp, Ann. Rev. Nucl. Sci. 15 (1965) 325. L. Wolfenstein, Phys. Rev. 96 (1954) 1654; Ann. Rev. Nucl. Sci. 6 (1956) 43. Y.S.Kim, Phys. Rev. 129 (1963) 862. L. D. Roper, Phys. Rev. 155 (1967) 1744. W. Rarita, R. J. Riddel, C. B. Chiu and R. J. N. Phillips, UCRL-17523 (1967)) unpublished. [7] Proc. Int. Conf. on polarized targets and ion sources, Saclay (1966). [8] H. P. Stapp, Phys. Rev. 103 (1956) 425. [9] G. C. Wick, Ann. of Phys. 18 (1962) 65. [lo] A. Biahs and B. E. Y. Svensson, Nuovo Cimento 42 (1965) 908. [ll] H. Steiner, UCRL-17903 (1967), unpublished. [12] P. du Val, Homographies , quaternions and rotations (Oxford University Press, 1964). 113) X. A. Tijrnqvist, Phys. Rev. 161 (1967) 1581; Nucl. Phys. B6 (1968) 187. [14] C. Lovelace, Proc. Heidelberg Int. Conf. on elementary particles, 1967 (NorthHolland Publ. Comp., Amsterdam, 1968); A. Donnachie , R. G. Kirsopp and C. Lovelace, Phys. Letters 26B (1968) 161.